A semicircle is drawn on AC. Two chords AB and BC of length 8 cm and 6 cm respectively are drawn in the semicircle. What is the measure of the diameter of the circle?a)14 cm.b)10 cm.c)12 cm.d)11 cm.Correct answer is option 'B'. Can you explain this answer?

Question

Answer

Since the point B lies on the circumference of the semicircle. Therefore, angle B=90 Using Pythagoras theorem, AB^2+BC^2=AC^2 => 8^2 + 6^2=AC^2 => 64+36=AC^2 => AC^2=100 => AC=10cm

Answer

The diameter of circle is AC. Here ∠ABC is angle of semicircle.

AB ∴ ∠ ABC = 90° ∴ Δ ABC is aright angled triangle.

10 cm

Answer

To solve this problem, we can start by drawing a diagram to represent the given information. Let's label the center of the semicircle as O, and the midpoint of chord AB as D.

1. Draw the semicircle:
- Draw a line segment AC.
- Mark point O as the center of the semicircle on AC.
- Draw a half-circle with O as the center and AC as the diameter.

2. Draw the chords:
- Draw chord AB with a length of 8 cm.
- Draw chord BC with a length of 6 cm.

3. Mark the midpoint:
- Find the midpoint of AB and mark it as point D.

Now, let's use the properties of a semicircle and chords to find the diameter of the circle.

4. Properties of a semicircle:
- The diameter of the semicircle is equal to the diameter of the circle.
- The radius of the semicircle is half the length of the diameter.

5. Properties of chords:
- The perpendicular bisector of a chord passes through the center of the circle.
- The perpendicular bisector of a chord divides the chord into two equal parts.

6. Using the properties:
- Since point O lies on the perpendicular bisector of chord AB, OD is the perpendicular bisector of AB.
- Therefore, OD is equal to half the length of AB, which is 4 cm.
- Similarly, OD is also equal to half the length of BC, which is 3 cm.
- Since OD is the radius of the semicircle, the diameter of the semicircle is twice the length of OD, which is 8 cm.

7. Conclusion:
- The diameter of the circle is equal to the diameter of the semicircle, which is 8 cm.

Therefore, the correct answer is option B) 10 cm.

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